Algebraic systems

Let ${\displaystyle F=(f_{1},\dots ,f_{k})=(p_{1}/q_{1},\dots ,p_{k}/q_{k})}$ be a finite set of rational functions over the field ${\displaystyle \mathbb {R} }$ where ${\displaystyle p_{i}}$ and ${\displaystyle q_{i}}$ are polynomials in ${\displaystyle \mathbb {R} [Z]}$ i.e. in variables ${\displaystyle Z=(z_{1},\dots ,z_{n})}$ with coefficients in ${\displaystyle \mathbb {R} }$. An algebraic system associated to ${\displaystyle F}$ is defined by the following equalities and inequalities:

${\displaystyle {\begin{array}{ccc}\left\{{\begin{array}{l}p_{1}(Z)=0,\\\vdots \\p_{k}(Z)=0\end{array}}\right.&{\mbox{and}}&\left\{{\begin{array}{l}q_{1}(Z)\neq 0,\\\vdots \\q_{k}(Z)\neq 0.\end{array}}\right.\end{array}}}$

An algebraic system defined by ${\displaystyle F=(f_{1},\dots ,f_{k})}$ is regular (a.k.a. smooth) if the system ${\displaystyle F}$ is of maximal rank ${\displaystyle k}$, meaning that the Jacobian matrix ${\displaystyle (\partial f_{i}/\partial z_{j})}$ is of rank ${\displaystyle k}$ at every solution ${\displaystyle Z}$ of the associated semi-algebraic variety.

Definition of Lie point symmetries

The following theorem (see th. 2.8 in ch.2 of ^[5]) gives necessary and sufficient conditions so that a local Lie group ${\displaystyle G}$ is a symmetry group of an algebraic system.

Theorem. Let ${\displaystyle G}$ be a connected local Lie group of a continuous dynamical system acting in the n-dimensional space ${\displaystyle \mathbb {R} ^{n}}$. Let ${\displaystyle F:\mathbb {R} ^{n}\rightarrow \mathbb {R} ^{k}}$ with ${\displaystyle k\leq n}$ define a regular system of algebraic equations:

${\displaystyle f_{i}(Z)=0\quad \forall i\in \{1,\dots ,k\}.}$

Then ${\displaystyle G}$ is a symmetry group of this algebraic system if, and only if,

${\displaystyle \delta f_{i}(Z)=0\quad \forall i\in \{1,\dots ,k\}{\mbox{ whenever }}f_{1}(Z)=\dots =f_{k}(Z)=0}$

for every infinitesimal generator ${\displaystyle \delta }$ in the Lie algebra ${\displaystyle {\mathfrak {g}}}$ of ${\displaystyle G}$.

Example

Consider the algebraic system defined on a space of 6 variables, namely ${\displaystyle Z=(P,Q,a,b,c,l)}$ with:

${\displaystyle \left\{{\begin{array}{l}f_{1}(Z)=(1-cP)+bQ+1,\\f_{2}(Z)=aP-lQ+1.\end{array}}\right.}$

The infinitesimal generator

${\displaystyle \delta =a(a-1){\dfrac {\partial }{\partial a}}+(l+b){\dfrac {\partial }{\partial b}}+(2ac-c){\dfrac {\partial }{\partial c}}+(-aP+P){\dfrac {\partial }{\partial P}}}$

is associated to one of the one-parameter symmetry groups. It acts on 4 variables, namely ${\displaystyle a,b,c}$ and ${\displaystyle P}$. One can easily verify that ${\displaystyle \delta f_{1}=f_{1}-f_{2}}$ and ${\displaystyle \delta f_{2}=0}$. Thus the relations ${\displaystyle \delta f_{1}=\delta f_{2}=0}$ are satisfied for any ${\displaystyle Z}$ in ${\displaystyle \mathbb {R} ^{6}}$ that vanishes the algebraic system.

Systems of ODEs and associated infinitesimal generators

Let ${\displaystyle d\cdot /dt}$ be a derivation w.r.t. the continuous independent variable ${\displaystyle t}$. We consider two sets ${\displaystyle X=(x_{1},\dots ,x_{k})}$ and ${\displaystyle \Theta =(\theta _{1},\dots ,\theta _{l})}$. The associated coordinate set is defined by ${\displaystyle Z=(z_{1},\dots ,z_{n})=(t,x_{1},\dots ,x_{k},\theta _{1},\dots ,\theta _{l})}$ and its cardinal is ${\displaystyle n=1+k+l}$. With these notations, a system of first-order ODEs is a system where:

${\displaystyle \left\{{\begin{array}{l}{\dfrac {dx_{i}}{dt}}=f_{i}(Z){\mbox{ with }}f_{i}\in \mathbb {R} (Z)\quad \forall i\in \{1,\dots ,k\},\\{\dfrac {d\theta _{j}}{dt}}=0\quad \forall j\in \{1,\dots ,l\}\end{array}}\right.}$

and the set ${\displaystyle F=(f_{1},\dots ,f_{k})}$ specifies the evolution of state variables of ODEs w.r.t. the independent variable. The elements of the set ${\displaystyle X}$ are called state variables, these of ${\displaystyle \Theta }$ parameters.

One can associate also a continuous dynamical system to a system of ODEs by resolving its equations.

An infinitesimal generator is a derivation that is closely related to systems of ODEs (more precisely to continuous dynamical systems). For the link between a system of ODEs, the associated vector field and the infinitesimal generator, see section 1.3 of.^[4] The infinitesimal generator ${\displaystyle \delta }$ associated to a system of ODEs, described as above, is defined with the same notations as follows:

${\displaystyle \delta ={\dfrac {\partial }{\partial t}}+\sum _{i=1}^{k}f_{i}(Z){\dfrac {\partial }{\partial x_{i}}}\cdot }$

Definition of Lie point symmetries

Here is a geometrical definition of such symmetries. Let ${\displaystyle {\mathcal {D}}}$ be a continuous dynamical system and ${\displaystyle \delta _{\mathcal {D}}}$ its infinitesimal generator. A continuous dynamical system ${\displaystyle {\mathcal {S}}}$ is a Lie point symmetry of ${\displaystyle {\mathcal {D}}}$ if, and only if, ${\displaystyle {\mathcal {S}}}$ sends every orbit of ${\displaystyle {\mathcal {D}}}$ to an orbit. Hence, the infinitesimal generator ${\displaystyle \delta _{\mathcal {S}}}$ satisfies the following relation^[8] based on Lie bracket:

${\displaystyle [\delta _{\mathcal {D}},\delta _{\mathcal {S}}]=\lambda \delta _{\mathcal {D}}}$

where ${\displaystyle \lambda }$ is any constant of ${\displaystyle \delta _{\mathcal {D}}}$ and ${\displaystyle \delta _{\mathcal {S}}}$ i.e. ${\displaystyle \delta _{\mathcal {D}}\lambda =\delta _{\mathcal {S}}\lambda =0}$. These generators are linearly independent.

One does not need the explicit formulas of ${\displaystyle {\mathcal {D}}}$ in order to compute the infinitesimal generators of its symmetries.